\newproblem{lay:7_2_7}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 7.2.7}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Make a change of variable, $\mathbf{x}=P\mathbf{y}$, that transforms the quadratic form $x_1^2+10x_1x_2+x_2^2$ into a quadratic form with no 
	cross-product term. Give $P$ and the new quadratic form.
}{
   % Solution
	If we orthogonally diagonalize the quadratic form, we obtain $A=PDP^T$
	\begin{center}
		$A=\begin{pmatrix}1 & 5 \\ 5 & 1 \end{pmatrix}=\begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}
		   \begin{pmatrix} 6 & 0 \\ 0 & -4 \end{pmatrix}
			 \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}^T$
	\end{center}
	We need to do the change of variables
	\begin{center}
		$\mathbf{x}=P\mathbf{y} \Rightarrow \mathbf{y}=P^T\mathbf{x}=\begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}
		   \begin{pmatrix} x_1\\x_2 \end{pmatrix}=\begin{pmatrix} \frac{1}{\sqrt{2}}(x_1+x_2) \\ \frac{1}{\sqrt{2}}(-x_1+x_2) \end{pmatrix}$
	\end{center}
	In this new set of variables, we have that the quadratic form is
	\begin{center}
		$Q(\mathbf{y})=\mathbf{y}^TD\mathbf{y}=6y_1^2-4y_2^2$
	\end{center}
}
\useproblem{lay:7_2_7}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}

